Fundamental theorem of linear algebra

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In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m × n matrix A and its singular value decomposition:

A=U\Sigma V^\mathrm{T}\

First, each matrix A \in \mathbf{R}^{m \times n} ( A has m rows and n columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace definition containing space dimension basis
column space, range or image \mathrm{im}(A) or \mathrm{range} (A) \mathbf{R}^m r (rank) The first r columns of U
nullspace or kernel \mathrm{ker}(A) or \mathrm{null} (A) \mathbf{R}^n n - r (nullity) The last (n - r) columns of V
row space or coimage \mathrm{im}(A^\mathrm{T}) or \mathrm{range} (A^\mathrm{T}) \mathbf{R}^n r (rank) The first r columns of V
left nullspace or cokernel \mathrm{ker}(A^\mathrm{T}) or \mathrm{null} (A^\mathrm{T}) \mathbf{R}^m m - r (corank) The last (m - r) columns of U

Secondly:

  1. In \mathbf{R}^n, \mathrm{ker}(A) = (\mathrm{im}(A^\mathrm{T}))^\perp, that is, the nullspace is the orthogonal complement of the row space
  2. In \mathbf{R}^m, \mathrm{ker}(A^\mathrm{T}) = (\mathrm{im}(A))^\perp, that is, the left nullspace is the orthogonal complement of the column space.
The four subspaces associated to a matrix A.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.

Further, all these spaces are intrinsically defined – they do not require a choice of basis – in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as A\colon V \to W and A^* \colon W^* \to V^*: the kernel and image of A^* are the cokernel and coimage of A.

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